In other words, the origins of the xp and xd axes are chosen such that at xp = xd = 0, internal engine pressure is equal to the external atmospheric pressure. This makes the zero state an equilibrium of the dynamics.

This follows from the observation that un + 1 − nn needs to be in the direction δu that maximizes (29) subject to the constraint of zero average. This direction is simply the projection of the square bracketed term onto the subspace of zero-average signals, i.e., removing the DC term.

In our particular implementation, all penalty functions (fixed and variable) are sums of reflections and shifts of a one-sided penalty function used as a basic building block.

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received August 28, 2013; final manuscript received December 22, 2014; published online March 4, 2015. Assoc. Editor: John B. Ferris.

Abstract

We consider an optimal control problem for a model of a Stirling engine that is actively controlled through its displacer piston motion. The framework of optimal periodic control (OPC) is used as the setting for this active control problem. We use the idealized isothermal Schmidt model for the system dynamics and formulate the control problem so as to maximize mechanical power output while trading off a penalty on control (displacer motion) effort. An iterative first-order algorithm is used to obtain the optimal periodic motion of the engine and control input. We show that optimal motion is typically nonsinusoidal with significant higher harmonic content, and that a significant increase in the power output of the engine is possible through the optimal scheduling of the displacer motion. These results indicate that OPC may provide a framework for a large class of energy conversion and harvesting problems in which active actuation is available.

References

Figures

Conceptual diagrams of Stirling engines where (a) shows the basic compartments and pistons, while (b) shows a beta-type engine where pistons are kinematically linked through a flywheel, and (c) shows the new concept of an engine with active control through direct actuation of the displacer piston. Linkages and actuators are shown conceptually and their actual geometry is not reflected in these diagrams.

A schematic of the fixed and variable penalties on the displacer (top) and power (bottom) pistons' positions. The variable penalties' shifts Sp, S¯d, and S¯d are parameters determined at each iteration step of the algorithm to enforce condition (40).

A plot showing the effect of three parameters of a beta engine on power output. The parameters are the phase difference, displacer, and piston amplitude. Small spheres represent small objective values while large dark gray spheres represent large objective values. The small dots represent points that either does not produce limit cycles or results in collisions. The optimal phase is around 90 deg, power piston amplitude has relatively little effect on performance, while larger displacer amplitudes produce more power.

The optimal motions and the pressure and velocity curves are displayed here for the actuated displacer model. The optimal displacer motion resembles that of a square wave. This maximizes the time spend at both pressure extremes.

The optimal motions and the pressure and velocity curves are displayed here for the beta Stirling model. The optimal piston motions resemble that of a sine wave. This is a result of the rotational inertia causing the flywheel to spin at near constant speed.

A PV diagram showing the optimally actuated cycle and the optimal beta cycle. The curves proceed clockwise and the area enclosed by the either curve is the mechanical energy output (per cycle) of the engine.

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